A.12 Answer: TW 12 tutorial

Answers for Sect. 12.2

No answer. But a line is fine for Plots 1, 2, 6 and 3 (but very weak!), but not for Plots 4 and 5.
My very rough slope estimates are:
Plot 1: $(50-10)/10 \approx 4$; Plot 2: $(20 - 50)/15 \approx -2$; Plot 3: $0$; Plot 6: $(55 - 35)/5 = 4$.
My very rough intercept estimates are: Plot 1: $8$; Plot 2: $40$; Plot 3: $32$; Plot 6: $10$.
My very rough estimates are: Plot 1; $\hat{y} = 8 + 4x$; Plot 2: $\hat{y} = 40 - 2x$; Plot 3: $\hat{y} = 32$; Plot 6: $\hat{y} = 10 + 4x$.

Answers for Sect. 12.3

Approximately linear, negative, reasonably strong.
Condition, extras (air con, etc.), sedan/hatch, colour, when rego due, new/old tyres, location, etc.
No answer.
Looks to be expensive, as $15,000 would be above the line (at least for the line I'd draw).
Probably $3900.
My guess is about $b_0\approx 17$ or $17,000. This would mean the average price of a 2014 second-hand Corolla can be expected to be about $17,000.
$b_1 = (1 - 17)/(16 - 0)\approx -1$. That is, the price reduces by about $1000 each year older the Corolla gets.
Using the above, we have $\hat{y} = 17 - x$ approximately. Guessing the regression line won't, of course, produce this level of precision, so anything close-ish to this is fine.
$\hat{y} = 16.54 - 0.96x$ (jamovi) or $\hat{y} = 16.536 - 0.963x$ (SPSS), where $y$ is the price in thousands of dollars, and $x$ is the age in years.
$r = -0.929$, and so $R^2 = (-0.929)^2 = 0.863$, or about $86$%, so about $86$% of the variation in prices can be explained by age alone. The rest can be explained by the car's condition, odometer reading, accessories, service history, etc.
jamovi: $\hat{y} = 16.54 - (0.96\times 20) = -2.66$, or -$2660.
SPSS: $\hat{y} = 16.536 - (0.963\times 20) = -2.72$, or -$2720.
This is clearly silly, as we are extrapolating.
Hardly need a test... $H_0$: $\beta_1 = 0$ vs $H_1$: $\beta_1 < 0$. From output, $t = -15.059$, and $P = 0.000/2 = 0.000$, so $P < 0.0001$: very strong evidence that older cars fetch lower second-hand prices, on average.
$-0.963 \pm (2\times 0.064)$, or $-0.963\pm 0.128$, or from $-1.091$ to $-0.835$.
Looks the same really, just reflected left-to-right.
- Size of $r$ won't change, sign from negative to positive; i.e., $r = 0.93$.
- Value of $R^2$ will be the same.
  - Slope will be the same except sign will change (in both cases, the values on the horizontal axis are one year apart).
- Intercept will change a lot... it is the predicted value of the price if the line is extended to year $0$ (which is, of course, meaningless).

Answers for Sect. 12.4

FIGURE A.1: Plots giving different correlations.